In this short appendix we analyze the vacuum structure of the 3d model \(\mathcal {N}=1\)  \(U(1)_{\frac{1}{2}}\)  with one charge-1 chiral \(Q\) . This model is well suited to understand the effects of a deformation with a linear monopole superpotential term \(\delta \mathcal {W}={\mathrm {Re}}({\mathfrak {M}})\) . This theory has a dual model which has been described in [1]}. The duality we are considering is
\( \begin{array}{c} U(1)^{\mathcal {N}=1}_{\frac{1}{2}} \, \text{with one chiral } Q \\ \mathcal {W}=-\frac{1}{4}{Q}^4 \end{array}\Longleftrightarrow \begin{array}{c} \text{WZ model} \, \text{with a real } H \text{ and a complex } P \text{ superfields} \\ \mathcal {W}= H{P}^2-\frac{H^3}{3} \end{array}\) 
